Length of Common Chord

Q. A rhombus is inscribed in the region common to the two circles $x^2+y^2-4x-12=0$ and $x^2+y^2+4x-12=0$ with two of its vertices on the line joining the centers of the circles. The area of the rhombus is

1. $6\sqrt{3}$ sq. units
2. $8\sqrt{3}$ sq. units
3. $4\sqrt{3}$ sq. units
4. $2\sqrt{3}$ sq. units

Q. Match the following by approximately matching the lists based on the information given in Column I and Column II

$\begin{array}{cc}Column1& Column2\\ \text{a. The length of the common chord of two circles of radii}3\text{and}& \text{p.}1\\ 4\text{units which intersect orthogonally is}\frac{k}{5},\text{then}k\text{is equal to}& \\ \text{b. The circumference of the circle}{x}^2+y^2+4x+12y+p=0& \text{q.}24\\ \text{is bisected by the circle}{x}^2+y^2-2x+8y-q=0\text{, then}& \\ p+q\text{is equal to}& \\ \text{c. Number of distinct chords of the circle}2x(x-\sqrt{2})+y(2y-1)& \text{r.}32\\ =0\text{chords are passing through the point}(\sqrt{2},\frac{1}{2})\text{and are}& \\ \text{bisected on x-axis is}& \\ \text{d. One of the diameters of the circle circumscribing the rectangle}& \text{s.}36\\ ABCD\text{is}4y=x+7\text{. If}A\text{and}B\text{are the points}(-3,4)\text{and}& \\ (5,4)\text{respectively,}\text{then the area of rectangle is}& \end{array}$

1. $a-q,b-s,c-p,d-r$
2. $a-p,b-s,c-q,d-r$
3. $a-r,b-s,c-p,d-q$
4. $a-q,b-r,c-p,d-s$

Q. The length of a common internal tangent to two circles is $7$ and a common external tangent is $11$. If the product of the radii of the two circles is $p$, then the value of $\frac{p}{2}$ is

Q. Circles $C_1$ & $C_2$ externally touch each other and they both internally touch another circle $C_3$. The radii of $C_1$ & $C_2$ are $4$ and $10$ respectiveley, and the centers of the three circles are collinear. A chord of $C_3$ is a transverse common tangent to the circles $C_1$ and $C_2$. Find the length of the chord

1. $14$
2. $4\sqrt{10}$
3. $8\sqrt{5}$
4. None of these

Q. Two circles with equal radii are intersecting at the points $(0,1)$ and $(0,-1)$. The tangent at the point $(0,1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

1. $1$
2. $2\sqrt{2}$
3. $2$
4. $\sqrt{2}$

Q. If the angle of intersection at a point where the two circles with radii $5cm$ and $12cm$ intersect is $90°,$ then the length $\left(\text{in}cm\right)$ of their common chord is :

1. $\frac{13}{5}$
2. $\frac{120}{13}$
3. $\frac{13}{2}$
4. $\frac{60}{13}$

Q. If the circles $x^2+y^2+5Kx+2y+K=0$ and $2(x^2+y^2)+2Kx+3y-1=0,(K\in R),$ intersect at the points $P$ and $Q$, then the line $4x+5y-K=0$ passes through $P$ and $Q$, for :

1. exactly one value of $K$
2. exactly two values of $K$
3. infinitely many values of $K$
4. no value of $K$

Q. Let any circle $S$ passes through the point of intersection of lines $\sqrt{3}(y-1)=x-1$ and $y-1=\sqrt{3}(x-1)$ and having its centre on the acute angle bisector of the given lines. If the common chord of $S$ and the circle $x^2+y^2+4x-6y+5=0$ passes through a fixed point, then the fixed point is

1. $(\frac{1}{3},\frac{3}{2})$
2. $(\frac{1}{2},\frac{3}{2})$
3. $(\frac{1}{2},\frac{3}{4})$
4. $(\frac{3}{2},\frac{3}{2})$

Q. For the given circles $x^2+y^2+2x+4y-20=0$ and $x^2+y^2+6x-8y+10=0$, which of the following is/are correct?

1. The number of common tangents is $2.$
2. The number of common tangents is $3.$
3. Length of the common tangent is $(1500{)}^{1/4}$ units
4. Length of the common tangent is $(1000{)}^{1/4}$ units

Q. $P(a,b)$ is a point in the first quadrant. Circles are drawn through $P$ touching the coordinate axes such that the length of common chord of these circles is maximum, if possible values of $a/b$ is $k_1$ and $k_2$, then $k_1+k_2$ is equal to